[R] Solution to differential equation

[Ravi Varadhan]

This integral is NOT easy.  Your solution is wrong.  You cannot integrate
term-by-term when the polynomial is in the *denominator* as it is here.

I am not sure if there is a closed-form solution to this.  I remember you
had posed a problem before that is only slightly different from this.  

Previous formulation:

dR/dt = k1*R*(1-(R/Rmax)^k2); R(0) = Ro

Current formulation:

dR/dt = k1*(R^k2)*(1-(R/Rmax)); R(0) = Ro
 

Note that the difference is that the exponent `k2' is in different places.
Initially I thought that this should not make much difference.  But
appearances of similarity can be quite deceiving.  Your previous formulation
admitted a closed-form solution, which I had shown you before.  But this
current formulation does not have a closed-form solution, at least not to my
abilities.

For the current formulation, you have u^k2 * (1-u) in the denominator of the
integrand. This cannot be simplified in terms of partial fractions.  In the
case of previous formulation, the denominator was u * (1 - u^k2), which
could be simplified as (1/u) + u^(k2-1)/(1-u^k2).  So, we are stuck, it
seems.

We can expand 1/(1 - u) in a power-series expansion, provided |x| < 1, and
then integrate each term of the expansion. However, this is not very helpful
as I don't know what this series converges to.

May be I am missing something simple here?  Any ideas?

Ravi.

-------------------------------------------------------
Ravi Varadhan, Ph.D.
Assistant Professor,
Division of Geriatric Medicine and Gerontology School of Medicine Johns
Hopkins University

Ph. (410) 502-2619
email: rvaradhan at jhmi.edu


-----Original Message-----
From: r-help-bounces at r-project.org [mailto:r-help-bounces at r-project.org] On
Behalf Of Scionforbai
Sent: Wednesday, December 15, 2010 12:28 PM
To: mahesh samtani
Cc: r-help at r-project.org
Subject: Re: [R] Solution to differential equation

> I am trying to find the analytical solution to this differential equation
>
> dR/dt = k1*(R^k2)*(1-(R/Rmax)); R(0) = Ro

> If there is an analytial solution to this differential equation then it

It is a polynomial function of R, so just develop the expression and
when you get the two terms in R (hint: one has exponent k2, the other
k2+1) you can simply apply the basic integration rule for polynomials.
It literally doesn't get easier than that.

the resulting function is:

k1/(k2+1) * R^(k2+1) - K1/[Rmax*(k2+2)] * R^(k2+2) + Ro

scion

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